Here \(\mathcal {S}_{jk}^{ab}\)  and \( \gamma ^{ab}_{jk}\)  is originated from the Fourier transform of
the two-point correlation function. They are Kramers-Kronig pairs to each-other.
Due to the common-environment effect, the atoms become entangled through the field correlation function.
This entanglement has an initial value dependency [1]}.
The field induced shift term is calculated from the Hilbert transforms of the response function of the field.
\(\mathcal {K}^{ab}(\omega _0) = \frac{\mathcal {P}}{\pi i}\int \limits ^{\infty }_{-\infty }d\omega \frac{\mathcal {G}^{ab}(\omega )}{\omega -\omega _0}~.\) 
